I want to compute the variance of this estimator $\hat{\sigma}^2 = \frac{n}{N}\sum_{i=1}^{N}\big(R_{i} - \frac{1}{N}\sum_{j=1}^{N}R_{j}\big)^2$, where $R_{1}, \ldots, R_{N}$ are i.i.d such that: $ R_{1}$ ~ $N(\frac{a}{n}, \frac{\sigma^2}{n})$. I already found its expectation which is $E[\hat{\sigma}^2] = \sigma^2 - \frac{\sigma^2}{N}$, but I am get stuck when it comes to calculate the variance, as I have to deal with correlated variables.
Any help highly appreciated! Thank you!